# Properties

 Label 304.4.a Level $304$ Weight $4$ Character orbit 304.a Rep. character $\chi_{304}(1,\cdot)$ Character field $\Q$ Dimension $27$ Newform subspaces $11$ Sturm bound $160$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 304.a (trivial) Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$160$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(304))$$.

Total New Old
Modular forms 126 27 99
Cusp forms 114 27 87
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$19$$FrickeDim
$$+$$$$+$$$+$$$8$$
$$+$$$$-$$$-$$$5$$
$$-$$$$+$$$-$$$7$$
$$-$$$$-$$$+$$$7$$
Plus space$$+$$$$15$$
Minus space$$-$$$$12$$

## Trace form

 $$27 q + 2 q^{5} + 14 q^{7} + 243 q^{9} + O(q^{10})$$ $$27 q + 2 q^{5} + 14 q^{7} + 243 q^{9} - 86 q^{11} - 46 q^{13} + 12 q^{15} - 26 q^{17} - 57 q^{19} + 136 q^{21} - 68 q^{23} + 865 q^{25} + 576 q^{27} - 398 q^{29} - 24 q^{33} - 246 q^{35} + 322 q^{37} - 540 q^{39} + 118 q^{41} - 230 q^{43} - 350 q^{45} + 1262 q^{47} + 915 q^{49} + 1012 q^{51} + 90 q^{53} + 830 q^{55} - 272 q^{59} - 390 q^{61} + 610 q^{63} - 348 q^{65} + 1020 q^{67} + 24 q^{69} + 1124 q^{71} - 1354 q^{73} + 1068 q^{75} + 1608 q^{77} + 1080 q^{79} + 1571 q^{81} - 1676 q^{83} - 1588 q^{85} - 3540 q^{87} - 314 q^{89} + 952 q^{91} - 2248 q^{93} + 760 q^{95} - 290 q^{97} + 1154 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(304))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 19
304.4.a.a $1$ $17.937$ $$\Q$$ None $$0$$ $$2$$ $$-9$$ $$31$$ $-$ $+$ $$q+2q^{3}-9q^{5}+31q^{7}-23q^{9}-57q^{11}+\cdots$$
304.4.a.b $1$ $17.937$ $$\Q$$ None $$0$$ $$5$$ $$-12$$ $$-11$$ $-$ $+$ $$q+5q^{3}-12q^{5}-11q^{7}-2q^{9}+54q^{11}+\cdots$$
304.4.a.c $2$ $17.937$ $$\Q(\sqrt{73})$$ None $$0$$ $$-9$$ $$-9$$ $$18$$ $-$ $+$ $$q+(-4-\beta )q^{3}+(-3-3\beta )q^{5}+(7+4\beta )q^{7}+\cdots$$
304.4.a.d $2$ $17.937$ $$\Q(\sqrt{177})$$ None $$0$$ $$-1$$ $$10$$ $$-57$$ $-$ $-$ $$q-\beta q^{3}+(6-2\beta )q^{5}+(-28-\beta )q^{7}+\cdots$$
304.4.a.e $2$ $17.937$ $$\Q(\sqrt{57})$$ None $$0$$ $$1$$ $$-5$$ $$6$$ $+$ $-$ $$q+\beta q^{3}+(-3+\beta )q^{5}+(5-4\beta )q^{7}+\cdots$$
304.4.a.f $2$ $17.937$ $$\Q(\sqrt{33})$$ None $$0$$ $$5$$ $$-5$$ $$30$$ $-$ $-$ $$q+(3-\beta )q^{3}-5\beta q^{5}+(17-4\beta )q^{7}+\cdots$$
304.4.a.g $3$ $17.937$ 3.3.7057.1 None $$0$$ $$-4$$ $$7$$ $$-7$$ $+$ $-$ $$q+(-1-\beta _{2})q^{3}+(2+2\beta _{1}+\beta _{2})q^{5}+\cdots$$
304.4.a.h $3$ $17.937$ 3.3.35529.1 None $$0$$ $$-1$$ $$9$$ $$-44$$ $-$ $+$ $$q+(-\beta _{1}-\beta _{2})q^{3}+(3+\beta _{2})q^{5}+(-15+\cdots)q^{7}+\cdots$$
304.4.a.i $3$ $17.937$ 3.3.3144.1 None $$0$$ $$-1$$ $$14$$ $$35$$ $-$ $-$ $$q-\beta _{2}q^{3}+(4-\beta _{1}+\beta _{2})q^{5}+(11-2\beta _{1}+\cdots)q^{7}+\cdots$$
304.4.a.j $3$ $17.937$ 3.3.3221.1 None $$0$$ $$5$$ $$2$$ $$35$$ $+$ $+$ $$q+(2+\beta _{2})q^{3}+(1-\beta _{1}+2\beta _{2})q^{5}+(11+\cdots)q^{7}+\cdots$$
304.4.a.k $5$ $17.937$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$0$$ $$-2$$ $$0$$ $$-22$$ $+$ $+$ $$q-\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-4-\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(304))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(304)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(38))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(76))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(152))$$$$^{\oplus 2}$$